Boundary Layer Theory — the Flow is Decided Near the Wall

The thin boundary layer on a flat plate, the friction velocity drawn out of wall shear stress, and the “law of the wall” for turbulent boundary layers — enough to answer why a CFD code asks for the $y^+$ of the first cell.

Opening

By the end of this chapter you should be able to do two things. First, sketch the law of the wall from scratch on a piece of paper. Second, explain to a friend why the line “first cell $y^+ \approx 1$” in a CFD report matters so much. The turbulence-viscosity models from Chapter 6 misbehave most stubbornly near walls, which is why the boundary layer comes up in almost every modelling discussion.

Main 1 — A boundary layer is a thin sheet stuck to the wall

The skin of a car, the wing of an airplane, the inside of a pipe — near every wall there is a thin region where viscosity cannot be ignored. We call this region the boundary layer. The velocity is zero at the wall itself (the no-slip condition) and approaches the free-stream value $U_\infty$ ($U$ infinity, free-stream velocity) as you move away from it.

For laminar flow over a flat plate, the Blasius solution gives an approximate boundary-layer thickness $\delta$ (delta, boundary layer thickness):

$$\delta(x) \approx 5 \sqrt{\nu x / U_\infty}$$

Here $x$ is the distance measured from the leading edge of the plate, and $\nu$ (nu, kinematic viscosity) is the kinematic viscosity. The message of the formula is simple — as the flow runs along the wall, the boundary layer thickens slowly, in proportion to $\sqrt{x}$.

Main 2 — Wall shear stress and friction velocity

Inside the boundary layer, the grip of the wall on the flow is captured by the wall shear stress $\tau_w$ (tau wall, wall shear):

$$\tau_w = \mu \left.\frac{\partial u}{\partial y}\right|_{y=0}$$

Here $\mu$ (mu, dynamic viscosity) is the dynamic viscosity, $u$ is the velocity along the wall, and $y$ is the perpendicular distance from the wall. The formula reads as “how steeply the velocity changes just above the wall, multiplied by viscosity.”

Divide $\tau_w$ by the density $\rho$ (rho, density) and take the square root, and you get a new velocity scale with units of m/s:

$$u_\tau = \sqrt{\tau_w / \rho}$$

We call $u_\tau$ the friction velocity. The name says “velocity,” but it is not the actual speed of any flow — it is a velocity-dimensioned quantity derived from the stress the wall exerts on the flow. If you re-measure every length and velocity inside the boundary layer in terms of $u_\tau$ and $\nu$, every flat-plate turbulent boundary layer collapses onto nearly the same shape. That is the punchline of the next section.

Main 3 — Wall units and the law of the wall

The non-dimensional versions of $y$ and $u$, scaled by the friction velocity and the kinematic viscosity, are called wall units:

$$y^+ = \frac{y u_\tau}{\nu}$$ $$u^+ = \frac{u}{u_\tau}$$

$y^+$ (y plus) is the dimensionless distance from the wall, $u^+$ (u plus) is the dimensionless velocity. Redrawn in these coordinates, the turbulent boundary layer separates into three regions regardless of the specific flow:

• Viscous sublayer: $y^+ < 5$. Viscosity dominates and velocity is proportional to distance — $u^+ = y^+$.
• Buffer layer: $5 < y^+ < 30$. Viscous and turbulent effects compete on roughly equal terms. There is no simple closed-form expression.
• Log layer: $y^+ > 30$. Velocity grows like the logarithm of distance — $u^+ = (1/\kappa) \ln y^+ + B$, with $\kappa \approx 0.41$ (kappa, Kármán’s constant) and $B \approx 5.0$.

Taken together, the three regions are the law of the wall. Industrial RANS calculations that aim to resolve the viscous sublayer demand first cell $y^+ \lesssim 1$. When the grid cost is too high, wall functions allow the first cell to sit at $30 \lesssim y^+ \lesssim 100$ and impose the log-law velocity there, at the price of accuracy for flows with separation or strong pressure gradients.

In Python

Let us draw the law of the wall directly. The viscous sublayer is a straight line, the log layer is a logarithmic curve, and the buffer layer is left intentionally blank with an annotation.

import numpy as np
import matplotlib.pyplot as plt

# Kármán constant and log-law intercept
kappa = 0.41
B = 5.0

# Viscous sublayer: y+ in [0.1, 5], u+ = y+
y_sub = np.linspace(0.1, 5.0, 50)
u_sub = y_sub

# Log layer: y+ in [30, 1000], u+ = (1/kappa) ln y+ + B
y_log = np.linspace(30.0, 1000.0, 200)
u_log = (1.0 / kappa) * np.log(y_log) + B

fig, ax = plt.subplots(figsize=(7, 5))
ax.semilogx(y_sub, u_sub, label=r"Viscous sublayer  $u^+ = y^+$")
ax.semilogx(y_log, u_log, label=r"Log layer  $u^+ = (1/\kappa)\ln y^+ + B$")

# Leave the buffer layer blank and annotate
ax.axvspan(5, 30, alpha=0.15, color="gray")
ax.text(12, 2, "Buffer layer\n(no closed form)", ha="center", va="center")

ax.set_xlabel(r"$y^+$")
ax.set_ylabel(r"$u^+$")
ax.set_title("Law of the wall — mean velocity profile in a turbulent boundary layer")
ax.legend()
ax.grid(True, which="both", alpha=0.3)
plt.tight_layout()
plt.show()

On a semilog x-axis the viscous sublayer looks like a curve and the log layer becomes a straight line — that is where the name “log layer” comes from. Saying the first cell has $y^+$ near one means that the mesh reaches all the way to the leftmost slice of this figure, the viscous sublayer.

To the next chapter

Chapter 8: Free shear flows moves to turbulence without walls — jets, wakes, and mixing layers. Once the wall is gone, the friction-velocity scale disappears with it, and the velocity difference and width of the flow itself become the new scales. Comparing the wall-bounded and free cases makes it clear, in reverse, why the wall units of Chapter 7 were such a special tool.